 Square-pulse resonance profile

A constant-amplitude drive lasting $T$ gives the Fourier transform in the title, where $\mathrm{sinc}x=\sin x/x$ and its value at zero is one. The transition probability in weak driving has a finite resonant maximum, first zeros at $\delta =\pm 2\pi /T$, and decreasing side lobes. Longer duration increases the peak as $T^2$ and narrows its width as $1/T$. The rotating-wave approximation discards a rapidly oscillating counterrotating term; it can have a small absolute error but a large relative error at the resonant term's zeros.